U.S. patent application number 15/547489 was filed with the patent office on 2019-03-28 for afm with suppressed parasitic signals.
The applicant listed for this patent is Ozgur Sahin. Invention is credited to Ozgur Sahin.
Application Number | 20190094265 15/547489 |
Document ID | / |
Family ID | 62978653 |
Filed Date | 2019-03-28 |
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United States Patent
Application |
20190094265 |
Kind Code |
A1 |
Sahin; Ozgur |
March 28, 2019 |
AFM with Suppressed Parasitic Signals
Abstract
An AFM that suppress parasitic deflection signals is described.
In particular, the AFM may use a cantilever with a probe tip that
is offset along a lateral direction from a longitudinal axis of
torsion of the cantilever. During AFM measurements, an actuator may
vary a distance between the sample and the probe tip along a
direction approximately perpendicular to a plane of the sample
stage in an intermittent contact mode. Then, a measurement circuit
may measure a lateral signal associated with a torsional mode of
the cantilever during the AFM measurements. This lateral signal may
correspond to a force between the sample and the probe tip.
Moreover, a feedback circuit may maintain, relative to a threshold
value: the force between the sample and the probe tip; and/or a
deflection of the cantilever corresponding to the force. Next, the
AFM may determine information about the sample based on the lateral
signal.
Inventors: |
Sahin; Ozgur; (New York,
NY) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Sahin; Ozgur |
New York |
NY |
US |
|
|
Family ID: |
62978653 |
Appl. No.: |
15/547489 |
Filed: |
January 26, 2017 |
PCT Filed: |
January 26, 2017 |
PCT NO: |
PCT/US17/15192 |
371 Date: |
July 29, 2017 |
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G01Q 10/065 20130101;
G01Q 60/30 20130101; G01Q 60/38 20130101; G01Q 70/10 20130101; G01Q
10/06 20130101 |
International
Class: |
G01Q 10/06 20060101
G01Q010/06; G01Q 60/38 20060101 G01Q060/38; G01Q 60/30 20060101
G01Q060/30 |
Claims
1. An atomic force microscope (AFM), comprising: a sample stage
configured to hold a sample; a cantilever with a probe tip, the
probe tip being offset along a lateral direction from a
longitudinal axis of torsion of the cantilever; a first actuator,
coupled to at least one of the sample stage and the cantilever,
configured to vary a distance between the sample and the probe tip
along a direction approximately perpendicular to a plane of the
sample stage in an intermittent contact mode; a measurement circuit
configured to measure a lateral signal associated with a torsional
mode of the cantilever during the AFM measurements, the lateral
signal corresponding to a force between the sample and the probe
tip, wherein, during the AFM measurements, the variation of the
distance has a fundamental frequency that is significantly less
than a lowest flexural resonance frequency of the cantilever; and a
feedback circuit, coupled to the measurement circuit and one of the
first actuator and a second actuator, configured to maintain,
relative to a threshold value, one of: the force between the sample
and the probe tip, and a deflection of the cantilever corresponding
to the force, wherein the second actuator is configured to change
the distance between the sample and the probe tip along the
direction, and wherein the AFM is configured to determine
information about the sample based on the lateral signal.
2. The AFM of claim 1, wherein the measurement circuit is
configured to measure a vertical signal associated with relative
displacement, along the direction, of the probe tip and the
sample.
3. The AFM of claim 1, wherein the AFM is further configured to
further determine the information based on the vertical signal.
4. The AFM of claim 1, wherein a contribution of parasitic signals
to the information is reduced without the AFM performing a recovery
operation; wherein the parasitic signals corresponding to phenomena
other than probe tip-sample interaction, thermal noise of the
cantilever and measurement-circuit noise; and wherein the recovery
operation involves performing measurements when the probe tip is
other than in contact with the sample.
5. The AFM of claim 1, wherein the information includes one of: the
force between the sample and the probe tip, topography of the
sample, and a material property of the sample.
6. The AFM of claim 1, wherein the feedback circuit is configured
to maintain one of: a peak force, an average force during a gating
interval, and a weighted average force during the gating
interval.
7. The AFM of claim 1, wherein the fundamental frequency is a
lesser of: the flexural resonance frequency divided by a square
root of two, and the flexural resonance frequency times one minus
an inverse of two times a quality factor of a flexural resonance of
the cantilever.
8. The AFM of claim 1, wherein the AFM further comprises: a
processor, coupled to the measurement circuit and at least one of
the first actuator and the second actuator, configured to execute a
program module; and memory, coupled to the processor, configured to
store the program module, wherein the program module, when executed
by the processor, causes the AFM to operate in the intermittent
contact mode and to determine the information.
9. The AFM of claim 1, wherein a ratio of an offset of the probe
tip along the lateral direction to a cantilever body length is
greater than or equal to 0.235 and a ratio of the offset to a
cantilever body lateral width is greater than or equal to 3.
10. The AFM of claim 1, wherein the first actuator is different
from the second actuator.
11. A method for determining information about a sample based on a
lateral signal, comprising: varying a distance between the sample
and a probe tip along a direction approximately perpendicular to a
plane of the sample in an intermittent contact mode, wherein the
probe tip is included in a cantilever and is offset along a lateral
direction from a longitudinal axis of torsion of the cantilever,
and wherein the variation of the distance has a lowest fundamental
frequency that is significantly less than a lowest flexural
resonance frequency of the cantilever; measuring the lateral signal
associated with a torsional mode of the cantilever during atomic
force microscopy (AFM) measurements, the lateral signal
corresponding to a force between the sample and the probe tip;
maintaining, using a feedback circuit in the AFM and relative to a
threshold value, one of: the force between the sample and the probe
tip, and a deflection of the cantilever corresponding to the force,
wherein maintaining the force involves changing the distance
between the sample and the probe tip along the direction, and
determining the information about the sample based on the lateral
signal.
12. The method of claim 11, wherein the method further comprises:
measuring a vertical signal associated with relative displacement,
along the direction, of the probe tip and the sample; and
determining the information is further based on the vertical
signal.
13. The method of claim 11, wherein a contribution of parasitic
signals to the information is reduced without performing a recovery
operation; wherein the parasitic signals corresponding to phenomena
other than probe tip-sample interaction and thermal noise of the
cantilever and noise associated with a detector that measures the
lateral signal; and wherein the recovery operation involves
performing measurements when the probe tip is other than in contact
with the sample.
14. The method of claim 11, wherein the feedback circuit maintains
one of: a peak force, an average force during a gating interval,
and a weighted average force during the gating interval.
15. The method of claim 11, wherein the fundamental frequency is a
lesser of: the flexural resonance frequency divided by a square
root of two, and the flexural resonance frequency times one minus
an inverse of two times a quality factor of a flexural resonance of
the cantilever.
16. A non-transitory computer-readable storage medium for use in
conjunction with an atomic force microscope (AFM), the
computer-readable storage medium configured to store a program
module that, when executed by the AFM, causes the AFM to: vary a
distance between a sample and a probe tip along a direction
approximately perpendicular to a plane of the sample in an
intermittent contact mode, wherein the probe tip is included in a
cantilever and is offset along a lateral direction from a
longitudinal axis of torsion of the cantilever, and wherein the
variation of the distance has a fundamental frequency that is
significantly less than a lowest flexural resonance frequency of
the cantilever; measure a lateral signal associated with a
torsional mode of the cantilever during atomic force microscopy
(AFM) measurements, the lateral signal corresponding to a force
between the sample and the probe tip; maintain, using a feedback
circuit in the AFM and relative to a threshold value, one of: the
force between the sample and the probe tip, and a deflection of the
cantilever corresponding to the force, wherein maintaining the
force involves changing the distance between the sample and the
probe tip along the direction, and determine information about the
sample based on the lateral signal.
17. The computer-readable storage medium of claim 16, wherein, when
executed by the AFM, the program module causes the AFM to: measure
a vertical signal associated with relative displacement, along the
direction, of the probe tip and the sample; and determine the
information further based on the vertical signal.
18. The computer-readable storage medium of claim 16, wherein a
contribution of parasitic signals to the information is reduced
without performing a recovery operation; wherein the parasitic
signals corresponding to phenomena other than probe tip-sample
interaction and thermal noise of the cantilever and noise
associated with a detector that measures the lateral signal; and
wherein the recovery operation involves performing measurements
when the probe tip is other than in contact with the sample.
19. The computer-readable storage medium of claim 16, wherein the
feedback circuit maintains one of: a peak force, an average force
during a gating interval, and a weighted average force during the
gating interval.
20. The computer-readable storage medium of claim 16, wherein the
fundamental frequency is a lesser of: the flexural resonance
frequency divided by a square root of two, and the flexural
resonance frequency times one minus an inverse of two times a
quality factor of the flexural resonance.
21. An atomic force microscope (AFM), comprising: a sample stage
configured to hold a sample; a cantilever with a probe tip, the
probe tip being offset along a lateral direction from a
longitudinal axis of torsion of the cantilever; an actuator,
coupled to at least one of the sample stage and the cantilever,
configured vary a distance between the sample and the probe tip
along a direction approximately perpendicular to a plane of the
sample stage in an intermittent contact mode; a measurement circuit
configured to measure a lateral signal associated with a torsional
mode of the cantilever during the AFM measurements, and configured
to measure a vertical signal associated with relative displacement,
along the direction, of the probe tip and the sample, wherein the
lateral signal corresponding to a force between the sample and the
probe tip, and wherein, during the AFM measurements, the variation
of the distance has a fundamental frequency that is significantly
less than a lowest flexural resonance frequency of the cantilever;
and wherein the AFM is configured to determine information about
the sample based on the lateral signal and the vertical signal.
22. The AFM of claim 21, wherein a contribution of parasitic
signals to the information is reduced without the AFM performing a
recovery operation; wherein the parasitic signals corresponding to
phenomena other than probe tip-sample interaction, thermal noise of
the cantilever and measurement-circuit noise; and wherein the
recovery operation involves performing measurements when the probe
tip is other than in contact with the sample.
23. The AFM of claim 21, wherein the information includes one of:
the force between the sample and the probe tip, topography of the
sample, and a material property of the sample.
24. The AFM of claim 21, wherein the fundamental frequency is a
lesser of: the flexural resonance frequency divided by a square
root of two, and the flexural resonance frequency times one minus
an inverse of two times a quality factor of the flexural
resonance.
25. The AFM of claim 21, wherein the AFM further comprises: a
processor, coupled to the measurement circuit and the actuator,
configured to execute a program module; and memory, coupled to the
processor, configured to store the program module, wherein the
program module, when executed by the processor, causes the AFM to
operate in the intermittent contact mode and to determine the
information.
26. The AFM of claim 21, wherein a ratio of an offset of the probe
tip along the lateral direction to a cantilever body length is
greater than or equal to 0.235.
27. The AFM of claim 21, wherein a ratio of the offset to a
cantilever body lateral width is greater than or equal to 3.
28. The AFM of claim 21, wherein the determination involves
correcting for parasitic signals in the lateral signal and the
vertical signal, the parasitic signals corresponding to phenomena
other than probe tip-sample interaction, thermal noise of the
cantilever and measurement-circuit noise.
29. A method for determining information about a sample,
comprising: varying a distance between the sample and a probe tip
along a direction approximately perpendicular to a plane of the
sample in an intermittent contact mode, wherein the probe tip is
included in a cantilever and is offset along a lateral direction
from a longitudinal axis of torsion of the cantilever, and wherein
the variation of the distance has a fundamental frequency that is
significantly less than a lowest flexural resonance frequency of
the cantilever; measuring a lateral signal associated with a
torsional mode of the cantilever during atomic force microscopy
(AFM) measurements and measuring a vertical signal associated with
relative displacement, along the direction, of the probe tip and
the sample, wherein the lateral signal corresponding to a force
between the sample and the probe tip; and determining the
information about the sample based on the lateral signal and the
vertical signal.
30. The method of claim 29, wherein a contribution of parasitic
signals to the information is reduced without performing a recovery
operation; wherein the parasitic signals corresponding to phenomena
other than probe tip-sample interaction and thermal noise of the
cantilever and noise associated with a detector that measures the
lateral signal; and wherein the recovery operation involves
performing measurements when the probe tip is other than in contact
with the sample.
31. The method of claim 29, wherein the information includes one
of: the force between the sample and the probe tip, topography of
the sample, and a material property of the sample.
32. The method of claim 29, wherein the fundamental frequency is a
lesser of: the flexural resonance frequency divided by a square
root of two, and the flexural resonance frequency times one minus
an inverse of two times a quality factor of the flexural
resonance.
33. The method of claim 29, wherein a ratio of an offset of the
probe tip along the lateral direction to a cantilever body length
is greater than or equal to 0.235; and wherein a ratio of the
offset to a cantilever body lateral width is greater than or equal
to 3.
34. The method of claim 29, wherein the determination involves
correcting for parasitic signals in the lateral signal and the
vertical signal, the parasitic signals corresponding to phenomena
other than probe tip-sample interaction and thermal noise of the
cantilever and noise associated with a detector that measures the
lateral signal.
35. A non-transitory computer-readable storage medium for use in
conjunction with an atomic force microscope (AFM), the
computer-readable storage medium configured to store a program
module that, when executed by the AFM, causes the AFM to: vary a
distance between a sample and a probe tip along a direction
approximately perpendicular to a plane of the sample in an
intermittent contact mode, wherein the probe tip is included in a
cantilever and is offset along a lateral direction from a
longitudinal axis of torsion of the cantilever, and wherein the
variation of the distance has a fundamental frequency that is
significantly less than a lowest flexural resonance frequency of
the cantilever; measure a lateral signal associated with a
torsional mode of the cantilever during atomic force microscopy
(AFM) measurements and measure a vertical signal associated with
relative displacement, along the direction, of the probe tip and
the sample, wherein the lateral signal corresponding to a force
between the sample and the probe tip; and determine information
about the sample based on the lateral signal and the vertical
signal.
36. The computer-readable storage medium of claim 35, wherein a
contribution of parasitic signals to the information is reduced
without performing a recovery operation; wherein the parasitic
signals corresponding to phenomena other than probe tip-sample
interaction and thermal noise of the cantilever and noise
associated with a detector that measures the lateral signal; and
wherein the recovery operation involves performing measurements
when the probe tip is other than in contact with the sample.
37. The computer-readable storage medium of claim 35, wherein the
information includes one of: the force between the sample, and the
probe tip, topography of the sample, and a material property of the
sample.
38. The computer-readable storage medium of claim 35, wherein the
fundamental frequency is a lesser of: the flexural resonance
frequency divided by a square root of two, and the flexural
resonance frequency times one minus an inverse of two times a
quality factor of the flexural resonance.
39. The computer-readable storage medium of claim 35, wherein a
ratio of an offset of the probe tip along the lateral direction to
a cantilever body length is greater than or equal to 0.235; and
wherein a ratio of the offset to a cantilever body lateral width is
greater than or equal to 3.
40. The computer-readable storage medium of claim 35, wherein the
determination involves correcting for parasitic signals in the
lateral signal and the vertical signal, the parasitic signals
corresponding to phenomena other than probe tip-sample interaction
and thermal noise of the cantilever and noise associated with a
detector that measures the lateral signal.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation of International
Application No. PCT/US2017/15192, "AFM with Suppressed Parasitic
Signals," by Ozgur Sahin, filed on Jan. 26, 2017, the contents of
which are herein incorporated by reference.
FIELD
[0002] The described embodiments relate to a technique for
performing Atomic Force microscopy (AFM) measurements with
suppressed parasitic deflection signals.
RELATED ART
[0003] Scanning probe microscopy encompasses a wide range of
imaging techniques. During these imaging techniques, a probe in a
scanning probe microscope (SPM) interacts with a sample to generate
a detectible signal that corresponds to or is indicative of the
interaction. In particular, the probe is often scanned across a
surface of the sample to generate images based on the detected
signals from the probe. Typically, the probes have very small
physical dimensions to improve the resolution of images. In
general, the images can reflect the topography and materials
properties that vary across the surface.
[0004] AFM is a special type of SPM that uses the mechanical
interaction of the probe with the sample. In an AFM, the probe
typically consists of a flexible cantilever beam (which is
sometimes referred to as a `cantilever`) with a sharp probe tip
placed on one end. Deflections of the cantilever can be indicative
of the forces between the probe tip and the sample. These
deflections are usually measured using a quadrant photo-detector
based on a laser beam that is reflected from the back of the
cantilever.
[0005] Because of their ability to obtain high-resolution images
under various environmental conditions (including ambient and
vacuum), AFMs have proven to be versatile imaging instruments. Some
AFMs generate an image of the sample with the probe tip in contact
with the sample surface. This imaging mode is commonly referred to
as a `contact mode`. In particular, during the contact mode, the
cantilever is typically brought in contact with the sample surface
and scanned across the sample. Then, a feedback system in the AFM
monitors the deflection of the cantilever and adjusts the relative
position of the cantilever with respect to the sample surface to
maintain a constant deflection (which is sometimes referred to as
`set point deflection`). The relative adjustment signals provided
by the feedback system can correspond to the surface topography,
which may be represented in images. Because the cantilever
deflection in the contact mode is proportional to the forces
exerted on the probe tip and the sample, lower deflection set
points are usually desirable in order to minimize damage to the
probe tip and the sample. However, if the deflection set point is
chosen too low, noise in the AFM detection system and drifts in the
measurement signals (e.g., because of thermally induced changes in
the cantilever shape) can prevent acquisition of images.
[0006] Contact-mode AFM also offers the possibility to
differentiate materials forming the sample based on differences in
their friction coefficients. For example, in lateral force
microscopy, which is based on contact-mode AFM, torsional
deflections of the cantilever caused by or associated with
frictional forces can be detected to create an image that provides
contrast indicative of the frictional characteristics of the sample
surface. Typically, the torsional deflection is detected from the
measurements signals provided by the quadrant photo-detector in the
AFM detection system, which can differentiate lateral and vertical
deflections of the cantilever. The lateral signal provided by the
quadrant photo-detector is usually sensitive to the torsional
deflections of the cantilever (around or about a longitudinal axis
of torsion) and the vertical signal at the quadrant photo-detector
is typically sensitive to the flexural deflections of the
cantilever. Because the cantilever is scanned across the surface
while the probe tip is in continuous contact with the sample,
frictional forces in contact-mode AFM can become significant and
they can damage the probe tip and the sample.
[0007] AFMs with intermittent contact modes, such as tapping-mode
AFM, largely overcome the limitation of the contact mode with
respect to probe tip-sample friction. For example, in tapping-mode
AFM, the cantilever is usually vibrated at or near its resonance
frequency and brought or placed proximate the sample surface so
that the vibrating probe tip makes intermittent contact with the
surface. The resulting intermittent interaction reduces the
probe-tip vibration amplitude. A feedback mechanism typically
adjusts the relative position of the cantilever with respect to the
surface in order to maintain the vibration amplitude at a
predetermined set-point value. Because the intermittent contact
reduces the frictional forces, which can reduce damage to the probe
tip and the sample, tapping-mode AFM has become among the most
popular AFM imaging modes.
[0008] In spite of the advantages of tapping-mode AFM, it can be
difficult to operate an AFM in the tapping mode because of the
non-linear dynamics of the vibrating cantilever. In general,
careful selection of the driving force, frequency, and set point
amplitude are typically needed in order to obtain good image
quality. One approach for addressing the challenges associated with
the tapping mode and the imaging process is peak-force tapping AFM.
In peak-force tapping AFM, the distance between the cantilever and
the sample is usually varied in an oscillatory or cyclical manner
(which is sometimes referred to as `Z-modulation`). During the
Z-modulation, the probe tip approaches, interacts, and retracts
from the surface. Moreover, in this process the flexural deflection
signals of the cantilever (which are provided by the quadrant
photo-detector) can allow substantially simultaneous determination
of the probe tip-sample forces. Because the probe tip-sample forces
are substantially instantaneously available with the probe-tip
deflection, AFMs operating in peak-force tapping mode can use the
peak value of the probe tip-sample force interactions in each
Z-modulation cycle to control the feedback loop in the feedback
system in order to track the topography of the sample surface.
[0009] However, because of parasitic deflection signals in the
peak-force tapping AFM, measured instantaneous deflection signals
often do not directly correspond to the instantaneous probe
tip-sample forces. Note that in the following discussion `parasitic
deflection signals` (which are sometimes referred to as `parasitic
signals`) are defined as the measurement signals associated with
operation of the AFM (e.g., cantilever deflection associated with
fluid drag, cantilever deflection associated with acceleration of
the cantilever during the Z-modulation, thermal noise of the
cantilever, and/or measurement-circuit noise, etc.). In order to
address or correct for the parasitic signals, many peak-force
tapping AFMs perform a so-called `recovery step.` During the
recovery step, the AFM usually determines and subtracts parasitic
deflection signals from the detected deflection signals in order to
obtain a deflection signal that is substantially free from
parasitic signals.
[0010] One recovery-step approach used for determining parasitic
signals in peak-force tapping AFM involves lifting the probe tip
away from the surface while turning the feedback off. By measuring
the background deflection signals in the absence of probe
tip-sample interactions, this approach can be used to determine the
parasitic signals and subtract the parasitic signals from the
detected deflection signals. Then, the feedback can be turned on
for imaging. While there are variations on this basic approach for
correcting the parasitic signals, the essential feature of this
correction technique is that the background parasitic deflection
signals are determined when the probe tip is lifted away from the
surface. However, the magnitude of some parasitic signals can
change depending on the position of the probe tip relative to the
surface. For example, the magnitude of viscous drag typically
varies with distance between the cantilever and the surface because
of squeezed-film effects. Moreover, velocity-dependent drag forces
usually change when the probe tip makes contact with the surface
because the probe-tip trajectory in time differs from a sinusoid.
In contrast, when the cantilever is lifted to prevent interaction
with the surface, the probe-tip trajectory in time is usually
sinusoidal. Furthermore, there are often long-range probe
tip-sample interaction forces, which the recovery step may
interpret as parasitic signals and, thus, which may be subtracted
from the deflection signals.
[0011] These effects often limit the accuracy of the recovery step
that is used to subtract parasitic deflection signals. Because one
of the primary advantages of peak-force tapping AFMs is the
improved feedback control based on peak probe tip-sample forces,
the inaccuracies in the recovery step can limit the potential of
peak-force tapping AFMs. For example, because of inaccurately
determined parasitic signals, the recovered probe tip-sample forces
can be larger or smaller than the actual probe tip-sample forces.
This error can limit the ability of a peak-force tapping AFM to
track the surface topography with low probe tip-sample forces.
Moreover, the inaccuracies can result in a loss of probe tip-sample
contact, or they can require the use of large forces that can
damage the probe tip and/or the sample. In addition, inaccurately
determined probe tip-sample force waveforms can reduce the ability
to determine and image material properties of the sample, because
these measurements often use force-distance curves determined from
probe tip-sample forces and probe tip-sample distances.
[0012] Consequently, the difficult in accurately determining the
parasitic deflection signals can degrade the measurements performed
using AFMs, and this can be frustrating to users.
SUMMARY
[0013] A first group of embodiments relate to an AFM. This AFM
includes: a sample stage that holds a sample; and a cantilever (or
an AFM cantilever) with a probe tip that is offset along a lateral
direction from a longitudinal axis of torsion of the cantilever.
Moreover, the AFM may include a first actuator that varies a
distance between the sample and the probe tip along a direction
approximately perpendicular to a plane of the sample stage in an
intermittent contact mode. Furthermore, the AFM may include a
measurement circuit that measures a lateral signal (which is
sometimes referred to as a `lateral deflection signal`) associated
with a torsional mode of the cantilever during the AFM
measurements, the lateral signal corresponding to a force between
the sample and the probe tip. Additionally, the AFM may include a
feedback circuit that maintains, relative to a threshold value: the
force between the sample and the probe tip; and/or a deflection of
the cantilever corresponding to the force. For example, the
feedback circuit may change the distance between the sample and the
probe tip along the direction using the first actuator and/or an
optional second actuator (which may be different from the first
actuator). Note that the AFM may determine information about the
sample based on the lateral signal.
[0014] In some embodiments, the measurement circuit measures a
vertical signal (which is sometimes referred to as a `vertical
deflection signal`) associated with relative displacement, along
the direction, of the probe tip and the sample.
[0015] Moreover, the AFM may further determine the information
based on the vertical signal.
[0016] Furthermore, a contribution of parasitic signals to the
information may be reduced without the AFM performing a recovery
operation or step, the parasitic signals may correspond to
phenomena other than probe tip-sample interaction, thermal noise of
the cantilever and measurement-circuit noise, and the recovery
operation may involve performing measurements when the probe tip is
other than in contact with the sample.
[0017] Additionally, the information may include: the force between
the sample and the probe tip, topography of the sample, and/or a
material property of the sample.
[0018] Note that the feedback circuit may maintain: a peak force,
an average force during a gating interval, and/or a weighted
average force during the gating interval.
[0019] In some embodiments, the variation of the distance has a
fundamental frequency that is less than a flexural resonance
frequency of the cantilever. For example, the fundamental frequency
may be less than a frequency that corresponds to the flexural
resonance frequency, such as significantly less than a lowest
flexural resonance frequency (and, thus, the AFM may not be
operated in a tapping mode, which may involve driving the
cantilever at or near a flexural resonance frequency and the use of
the vibration amplitude as the feedback signal). Note that the
fundamental frequency may be a lesser of: the flexural resonance
frequency divided by a square root of two, and the flexural
resonance frequency times one minus an inverse of two times a
quality factor of the flexural resonance. However, in some
embodiments, the fundamental frequency equals or is proximate to
the flexural resonance frequency, and the AFM may operate in a
tapping mode, however, with the feedback signal based on the peak
force or an instantaneous force. In these embodiments, the feedback
signal may be derived or determined from the lateral signal.
[0020] Moreover, the AFM may include: a processor that executes a
program module; and memory that stores the program module. When
executed by the processor, the program module causes the AFM to
operate in the intermittent contact mode and to determine the
information.
[0021] Furthermore, a ratio of an offset of the probe tip along the
lateral direction to a cantilever body length may be greater than
or equal to 0.235 and/or a ratio of the offset to a cantilever body
lateral width may be greater than or equal to 3.
[0022] Another embodiment provides a method for determining
information about the sample based on the lateral signal, which may
be performed by the AFM.
[0023] Another embodiment provides a computer-readable storage
medium that stores a program module for use with the AFM When
executed by the AFM, the program module causes the AFM to perform
at least some of the aforementioned operations.
[0024] A second group of embodiments relate to an AFM. This AFM
includes: a sample stage that holds a sample; and a cantilever (or
an AFM cantilever) with a probe tip that is offset along a lateral
direction from a longitudinal axis of torsion of the cantilever.
Moreover, the AFM may include an actuator that varies a distance
between the sample and the probe tip along a direction
approximately perpendicular to a plane of the sample stage in an
intermittent contact mode. Furthermore, the AFM may include a
measurement circuit that measures a lateral signal associated with
a torsional mode of the cantilever during the AFM measurements, and
that measures a vertical signal associated with relative
displacement, along the direction, of the probe tip and the sample.
Note that the lateral signal may correspond to a force between the
sample and the probe tip. Note that the AFM may determine
information about the sample based on the lateral signal and the
vertical signal.
[0025] Moreover, a contribution of parasitic signals to the
information may be reduced without the AFM performing a recovery
operation or step, the parasitic signals may correspond to
phenomena other than probe tip-sample interaction, thermal noise of
the cantilever and measurement-circuit noise, and the recovery
operation may involve performing measurements when the probe tip is
other than in contact with the sample.
[0026] Furthermore, the information may include: the force between
the sample and the probe tip, topography of the sample, and/or a
material property of the sample.
[0027] Additionally, the variation of the distance may have a
fundamental frequency that is less than a flexural resonance
frequency of the cantilever. For example, the fundamental frequency
may be less than a frequency that corresponds to the flexural
resonance frequency, such as significantly less than a lowest
flexural resonance frequency (and, thus, the AFM is not operated in
a tapping mode, which may involve driving the cantilever at or near
a flexural resonance frequency and the use of the vibration
amplitude as the feedback signal). Note that the fundamental
frequency may be a lesser of: the flexural resonance frequency
divided by a square root of two, and the flexural resonance
frequency times one minus an inverse of two times a quality factor
of the flexural resonance. However, in some embodiments, the
fundamental frequency equals or is proximate to the flexural
resonance frequency, and the AFM may operate in a tapping mode,
however, with the feedback signal based on the peak force or an
instantaneous force.
[0028] In some embodiments, the AFM includes: a processor that
executes a program module; and memory that stores the program
module. When executed by the processor, the program module causes
the AFM to operate in the intermittent contact mode and to
determine the information.
[0029] Note that a ratio of an offset of the probe tip along the
lateral direction to a cantilever body length may be greater than
or equal to 0.235 and/or a ratio of the offset to a cantilever body
lateral width may be greater than or equal to 3.
[0030] Moreover, the determination may involve correcting for
parasitic signals in the lateral signal and the vertical signal,
the parasitic signals corresponding to phenomena other than probe
tip-sample interaction, thermal noise of the cantilever and
measurement-circuit noise.
[0031] Another embodiment provides a method for determining
information about the sample, which may be performed by the
AFM.
[0032] Another embodiment provides a computer-readable storage
medium that stores a program module for use with the AFM When
executed by the AFM, the program module causes the AFM to perform
at least some of the aforementioned operations.
[0033] A third group of embodiments provides an AFM cantilever for
use with an AFM. This AFM cantilever includes: a cantilever body;
and a probe tip that is offset along a lateral direction from a
longitudinal axis of torsion of the cantilever. Moreover, the AFM
cantilever may have a torsional mode that suppresses parasitic
signals, the parasitic signals corresponding to phenomena other
than probe tip-sample interaction, thermal noise of the cantilever
and measurement-circuit noise.
[0034] A fourth group of embodiments relates to an electronic
device for use with an AFM. This electronic device includes first
input nodes that couple to a measurement circuit in the AFM and
that receive, from the measurement circuit, a measurement signal,
where the measurement signal includes a lateral signal associated
with a torsional mode of a cantilever in the AFM during AFM
measurements, and the lateral signal corresponds to a force between
a sample and a probe tip in the cantilever. Moreover, the
electronic device includes second input nodes that couple to a
feedback circuit in the AFM and that receive, from the feedback
circuit, a feedback signal, where the feedback signal corresponds
to a vertical signal associated with relative displacement, along a
direction approximately perpendicular to a plane of the sample, of
the probe tip and the sample. Furthermore, the electronic device
includes a signal-conditioning circuit that modifies the feedback
signal so that the modified signal corresponds to a force between
the sample and the probe tip. Additionally, the electronic device
includes first output nodes that couple to the feedback circuit and
that provide the measurement signal to the feedback circuit, and
second output nodes that couple to the measurement circuit and that
provide the modified feedback signal to the measurement
circuit.
[0035] Note that the signal-condition circuit may apply a
feed-forward modification to the feedback signal.
[0036] Another embodiment provides a method for modifying a
feedback signal, which may be performed by the electronic
device.
[0037] Another embodiment provides a computer-readable storage
medium that stores a program module for use with the electronic
device. When executed by the electronic device, the program module
causes the electronic device to perform at least some of the
aforementioned operations.
[0038] The preceding summary is provided as an overview of some
exemplary embodiments and to provide a basic understanding of
aspects of the subject matter described herein. Accordingly, the
above-described features are merely examples and should not be
construed as narrowing the scope or spirit of the subject matter
described herein in any way. Other features, aspects, and
advantages of the subject matter described herein will become
apparent from the following Detailed Description, Figures, and
Claims.
BRIEF DESCRIPTION OF THE FIGURES
[0039] FIG. 1 is a block diagram illustrating an example of an
atomic force microscope (AFM) in accordance with an embodiment of
the present disclosure.
[0040] FIG. 2 is a drawing illustrating an example of Z-modulation,
a vertical signal and a probe tip-sample force waveform as a
function of time in accordance with an embodiment of the present
disclosure.
[0041] FIG. 3A is a drawing illustrating an example of a cantilever
for use with the AFM of FIG. 1 in accordance with an embodiment of
the present disclosure.
[0042] FIG. 3B is a drawing illustrating an example of a cantilever
for use with the AFM of FIG. 1 in accordance with an embodiment of
the present disclosure.
[0043] FIG. 4 is a drawing illustrating an example of vertical
signal and parasitic-suppressed lateral signal measurements using
the AFM of FIG. 1 in accordance with an embodiment of the present
disclosure.
[0044] FIG. 5A is a drawing illustrating an example of a vertical
signal as a function of time in accordance with an embodiment of
the present disclosure.
[0045] FIG. 5B is a drawing illustrating an example of a lateral
signal as a function of time in accordance with an embodiment of
the present disclosure.
[0046] FIG. 6 is a drawing illustrating an example a cantilever for
use with the AFM of FIG. 1 in accordance with an embodiment of the
present disclosure.
[0047] FIG. 7 is a block diagram illustrating an example of a
parasitic lateral-signal estimation circuit for use with the AFM of
FIG. 1 in accordance with an embodiment of the present
disclosure.
[0048] FIG. 8 is a drawing illustrating an example of Z-modulation,
a vertical signal and a parasitic-suppressed lateral signal as a
function of time in accordance with an embodiment of the present
disclosure.
[0049] FIG. 9 is a drawing illustrating an example of a model of
parasitic flexural deflections during Z-modulation of a cantilever
in accordance with an embodiment of the present disclosure.
[0050] FIG. 10 is a flow diagram illustrating an example of a
method for determining information about a sample using the AFM of
FIG. 1 in accordance with an embodiment of the present
disclosure.
[0051] FIG. 11 is a flow diagram illustrating an example of a
method for determining information about a sample using the AFM of
FIG. 1 in accordance with an embodiment of the present
disclosure.
[0052] FIG. 12 is a block diagram illustrating an example of an
electronic device for use with an AFM in accordance with an
embodiment of the present disclosure.
[0053] FIG. 13 is a flow diagram illustrating an example of a
method for modifying a feedback signal using the electronic device
of FIG. 12 in accordance with an embodiment of the present
disclosure.
[0054] FIG. 14 is a block diagram illustrating an example of an
electronic device in accordance with an embodiment of the present
disclosure.
[0055] Note that like reference numerals refer to corresponding
parts throughout the drawings. Moreover, multiple instances of the
same part are designated by a common prefix separated from an
instance number by a dash.
DETAILED DESCRIPTION
[0056] In a first group of embodiments, an AFM that suppress
parasitic deflection signals is described. In particular, the AFM
may use a cantilever with a probe tip that is offset along a
lateral direction from a longitudinal axis of torsion of the
cantilever. During AFM measurements, an actuator may vary a
distance between the sample and the probe tip along a direction
approximately perpendicular to a plane of the sample stage in an
intermittent contact mode. Then, a measurement circuit may measure
a lateral signal associated with a torsional mode of the cantilever
during the AFM measurements. This lateral signal may correspond to
a force between the sample and the probe tip. Moreover, a feedback
circuit may maintain, relative to a threshold value: the force
between the sample and the probe tip; and/or a deflection of the
cantilever corresponding to the force. Next, the AFM may determine
information about the sample based on the lateral signal.
[0057] In a second group of embodiments, an AFM that suppress
parasitic deflection signals is described. In particular, the AFM
may use a cantilever with a probe tip that is offset along a
lateral direction from a longitudinal axis of torsion of the
cantilever. During AFM measurements, an actuator may vary a
distance between the sample and the probe tip along a direction
approximately perpendicular to a plane of the sample stage in an
intermittent contact mode. Then, a measurement circuit may measure
a lateral signal associated with a torsional mode of the cantilever
during the AFM measurements, and may measure a vertical signal
associated with relative displacement, along the direction, of the
probe tip and the sample. The lateral signal may correspond to a
force between the sample and the probe tip. Next, the AFM may
determine information about the sample based on the lateral signal
and the vertical signal.
[0058] By suppressing the parasitic deflection signals, the
measurement technique may improve probe tip-sample force
measurement and control during AFM measurements (such as those that
use peak force-based feedback). This capability may allow higher
Z-modulation fundamental frequencies and, thus, faster imaging
speeds (e.g., up to 10.times. faster) and improved image quality.
For example, in AFMs that rely on instantaneous probe tip-sample
forces for feedback (such as the magnitude of peak forces),
suppressing the parasitic deflection signals may allow topographic
imaging with lower forces (such as lower peak forces). Moreover,
because various sources of parasitic deflections (such as parasitic
deflections due to viscous drag forces and accelerations) depend on
the vertical oscillation speed of the cantilever, by suppressing
the parasitic deflection signals an AFM can tolerate faster
oscillation speeds. Consequently, the measurement technique may
allow larger Z-modulation fundamental frequencies and/or
oscillation amplitudes than existing AFM measurement techniques
with reduced parasitic signals. (Note that typical Z-modulation
amplitudes may be between 5 and 200 nm, but larger and smaller
amplitudes can also be used.) The resulting shorter oscillation
periods may reduce the feedback delay. Therefore, suppressing the
parasitic deflection signals can be used to improve the imaging
speed, i.e., to achieve a faster scan speed or a faster tip-sample
engagement process while keeping the tip-sample forces low.
Furthermore, suppressing the parasitic deflection signals may
eliminate the need for a recovery step or operation to determine
probe tip-sample forces. Additionally, the material properties and
topology of the sample may be more accurately determined. For
example, suppressing the parasitic deflection signals improves the
accurate probe tip-sample force waveforms, which can improve the
accuracy of mechanical property measurements based on
force-distance curves. Consequently, the measurement technique may
provide more flexible and accurate measurements, and may improve
the user experience when using the AFM.
[0059] We now describe embodiments of an AFM. FIG. 1 presents a
block diagram illustrating an example of an AFM 100. This AFM may
include: a sample stage 110 that holds a sample 112; a cantilever
114 with a probe tip 116 that is offset along a lateral direction
from a longitudinal axis of torsion of the cantilever 114 (e.g., by
more than 20 .mu.m); an actuator 118 (such as a piezoelectric
element), coupled to sample stage 110 and/or cantilever 114; and a
measurement circuit 124 (including a quadrant detector); an
optional feedback circuit 126 coupled to measurement circuit 124;
and/or an optional actuator 128 coupled to sample stage 110 and/or
cantilever 114. Note that optional feedback circuit 126 may be
coupled to actuator 118 and/or an optional actuator 128. AFM 100
may be used to perform measurements on a wide variety of samples,
including: a biological sample, a polymer, a gel, a thin film, a
patterned wafer, a data-storage device, an organic material, and/or
an inorganic material. Moreover, the measurements may be performed
in ambient, liquid, aqueous buffers, or vacuum.
[0060] As described further below with reference to FIG. 14, AFM
100 may include subsystems, such as a networking subsystem, a
memory subsystem and a processor subsystem. For example, memory
subsystem may store a program module that, when executed by the
processor subsystem, causes AFM 100 to perform the measurement
technique, which is described further below with reference to FIGS.
2-11. However, as described further below with reference to FIGS.
12-13, in some embodiments AFM 100 is used in conjunction with an
electronic device (which is sometimes referred to as an `instrument
module`), which facilitates the measurement technique.
[0061] As discussed previously, parasitic deflection signals, e.g.,
in peak-force tapping AFM c(and, more generally, when Z-modulation
is used, rather than driving the cantilever into flexural
resonance, and when the feedback relies on peak force, rather than
oscillation amplitude), can corrupt measurements of instantaneous
deflection signals. Consequently, the instantaneous deflection
signals may not directly correspond to the instantaneous probe
tip-sample forces.
[0062] In order to address this problem, actuator 118 may vary a
distance 120, such that associated with translational motion,
between sample 112 and probe tip 116 along a direction 122
approximately perpendicular (such as within 15.degree. of
perpendicular) to a plane of sample stage 110 in an intermittent
contact mode. (In addition, actuator 118 and/or a separate scanner,
not shown, may scan probe tip 116 in a plane of sample 112 or along
a surface of sample 112 to generate an image.) For example,
actuator 118 may vary a distance 120 or impart translational motion
between sample 112 and cantilever 114 along a direction 122 using
Z-modulation (such as with modulation fundamental frequencies
between 250 and 1 kHz, 10 kHz or 20 kHz, and more generally using
modulation fundamental frequencies that are typically below the
lowest fundamental flexural resonance frequency of cantilever 114),
so that probe tip 116 approaches, interacts with and moves away
from the surface of sample 112. (Note that this may be in contrast
with tapping mode, in which probe tip 116 is moved by exciting a
flexural motion of cantilever 114. However, in general during the
measurement technique distance 120 may vary because of
translational motion and/or excitation of an excitation mode of
cantilever 114.) In some embodiments, the variation of distance 120
has a fundamental frequency that is less than a flexural resonance
frequency of cantilever 114, such as significantly less than a
lowest flexural resonance frequency (i.e., cantilever 114 is
operated off-resonance and, thus, AFM 100 may not be operated in a
tapping mode, which typically involves driving the cantilever at or
near a flexural resonance frequency and the use of the vibration
amplitude as the feedback signal). Alternatively or additionally,
the fundamental frequency may be less than a frequency that
corresponds to the flexural resonance frequency. However, in some
embodiments, the fundamental frequency equals or is proximate to
the flexural resonance frequency, and thus AFM 100 may be operated
in tapping mode, however, with the feedback signal based on the
peak force or an instantaneous force. Note that a torsional
resonance frequency of cantilever 114 may be between 50 kHz and
several MHz.
[0063] Then, measurement circuit 124 may measure at least a
torsional deflection signal associated with a torsional mode of
cantilever 114 during the AFM measurements. For example, if
measurement circuit 124 includes a laser and a position-sensitive
quadrant photo-detector for deflection detection of cantilever 114,
measurement circuit 124 may measure at least a lateral signal
associated with a torsional mode of cantilever 114 during the AFM
measurements, where the lateral signal corresponds to a force
between sample 112 and probe tip 116. (Note that the measured
lateral signal in FIG. 1 is distinct from a so-called `lateral
force mode` in which an AFM detects torsional motion associated
with probe tip-sample frictional forces or plane forces. In the
measurement technique, the torsional deflections of cantilever 114
are due to vertical forces and occur because probe tip 116 is
offset along the lateral direction.)
[0064] Moreover, optional feedback circuit 126 (such as a
proportional-integral controller) may maintain, relative to a
threshold value (which is sometimes referred to as `a set point
value`): the force between sample 112 and probe tip 116, and/or a
deflection of cantilever 114 corresponding to the force. For
example, optional feedback circuit 126 may maintain: a peak force
as probe tip 116 interacts with sample 112, an average force during
a gating interval as probe tip 116 interacts with sample 112,
and/or a weighted average force during the gating interval as probe
tip 116 interacts with sample 112. Furthermore, the peak forces,
average forces during the gating interval, and/or the weighted
average force during the gating interval could be synchronously
averaged over many cycles of the fundamental frequency of the tip
oscillation. Alternatively, the peak forces, average forces during
the gating interval, and/or the weighted average force may be
determined from a synchronously averaged tip-sample force waveform
at the fundamental frequency of the tip oscillation. The feedback
may involve optional feedback circuit 126, using actuator 118
and/or optional actuator 128, changing distance 120 between sample
112 and probe tip 116 along direction 122.
[0065] In some embodiments, optional feedback circuit 126 compares
a vertical signal from measurement circuit 124 to a threshold. The
resulting difference may be input to a proportional control, which
outputs a feedback signal to actuator 118 and/or optional actuator
128. In general, the feedback may be based on the force and/or the
deflection measured vertical signal. Thus, the feedback signal may
correspond to or may be a function of the force and/or the
deflection.
[0066] As discussed further below, optional feedback circuit 126
may use the peak-force values (without or with reduced parasitic
signals) determined from torsional deflection signals to control a
feedback loop, e.g., to maintain a constant peak force at each
cycle of Z-modulation and to track the surface topography while
scanning probe tip 116 over sample 112. Thus, this measurement
technique may maintain a steady state interaction by comparing the
peak-force value to the set-point value and adjusting the relative
distance 120 between probe tip 116 and sample 112 based on the
comparison, thereby tracking the surface of sample 112 during the
scanning process.
[0067] Note that, based on the torsional deflection signal or the
lateral signal, a contribution of parasitic signals to the
peak-force values may be reduced or eliminated without AFM 100
performing a recovery operation or step, i.e., without determining
and subtracting the parasitic signals. The parasitic signals may
correspond to phenomena other than probe tip-sample interaction,
thermal noise of cantilever 114 and measurement-circuit noise, and
the recovery operation, which is typically performed in existing
AFMs but may not be performed in the measurement technique, may
involve performing measurements when probe tip 116 is other than in
contact with sample 112 in order to determine the parasitic signals
or the contribution of the parasitic signals. Consequently, the
measurement technique is sometimes referred to as a
`parasitic-suppressed AFM mode` or a `psAFM mode`. In addition, the
torsional deflection signal or the lateral signal if a quadrant
photo-detector is used in the psAFM mode are sometimes,
respectively, referred to as `parasitic-suppressed torsional
deflection signals` or `parasitic-suppressed lateral signals.`
Therefore, this measurement technique may incorporate the
advantages of using peak probe tip-sample interaction forces
without the limitations of other AFM measurement techniques and
without performing the recovery operation or step. Note that the
measurement technique may be used for imaging with low forces (such
as between 5 pN and 10 nN). However, the measurement technique may
be used with larger imaging forces, e.g., as large as 1 .mu.N.
[0068] Instead of performing the recovery step or operation, AFM
100 may use a baseline torsional deflection value (or lateral
deflection value) to adjust torsional deflection signals (or
lateral signals). For example, because of misalignments in a laser
position on a photo-detector in measurement circuit 124, there is
typically a non-zero, baseline lateral signal. This baseline
lateral signal value can be subtracted from measured lateral signal
so that lateral signals at the baseline are interpreted as zero
deflection and zero force. This may allow long-term imaging with
low peak probe tip-sample forces. Moreover, because of drifts in
AFM 100 (such as thermal drifts that cause cantilever 114 to bend
and twist), the baseline can gradually and slowly change over time.
Therefore, in some embodiments, AFM 100 determines the baseline
lateral signal repeatedly (such as periodically or after a time
interval) to readjust the detected parasitic-suppressed lateral
signals. Note that the baseline lateral signal value can be
determined from the value of the lateral signal when probe tip 116
is not interacting with the surface of sample 112. However, this
can be performed during normal operation of AFM 100, as opposed to
intentionally retracting probe tip 116 away from the surface, as is
typically the case in the recovery step or operation.
[0069] (While the measurement technique allows the parasitic
signals to be reduced or eliminated without the use of the recovery
step or operation, in some embodiments the measurement technique
includes a residual recovery step or operation to estimate and
subtract residual parasitic signals in the parasitic-suppressed
torsional deflection signals. Because these residual parasitic
signals are small in magnitude, uncertainties in their
determination can result in even smaller inaccuracies in the final
deflection waveform. For example, AFM 100 may: lift probe tip 116
away from the surface of sample 112, measure the background signal,
synthesize the background signal and subtract it from the
parasitic-suppressed torsional deflection signals.)
[0070] Furthermore, AFM 100 may determine information about sample
112 based on the lateral signal. For example, the information may
include: the force between sample 112 and probe tip 116, topography
of sample 112, and/or a material property of sample 112 (such as an
elastic modulus, a stiffness, a work of adhesion, a peak adhesive
force, another adhesion metric, a loss modulus, a storage modulus,
a hardness, an electrical property, an optical property, etc.). As
discussed above, by reducing or eliminating the contribution of the
parasitic signals, AFM 100 may determine the informationwithout AFM
100 performing the recovery operation or step.
[0071] In order to determine the electrical property or
characteristic, AFM 100 may apply a DC or AC voltage signal to
probe tip 116 relative to sample 112, so that the measured forces
from the parasitic-suppressed torsional deflection signals contain
information about the electrical properties of the surface of
sample 112, which can be used to determine electrical properties of
the surface. By scanning cantilever 114 across the surface of
sample 112, these measurements can be used to generate images that
map electrical properties of materials, such as the dielectric
constant, the resistivity, the electrical impedance, etc.
[0072] Moreover, in order to determine an optical property or
characteristic, AFM 100 may apply an electromagnetic pulse, an
infrared pulse or an optical pulse to probe tip 116 and/or sample
112, so that rapid topographic changes associated with absorption
of the pulse can be detected from the parasitic-suppressed
torsional deflections. By suppressing parasitic signals, the
measurement technique may improve the ability of AFM 100 to detect
small and rapid changes in surface height in response to the
absorption. This is because the variations in the surface height
may result in a change in the probe tip-sample force, which may
cause a change in the parasitic-suppressed torsional deflection
signal. Furthermore, because the absorption characteristics of
sample 112 depends on its chemical composition, the measurement
technique may improve the ability to detect chemical changes and
also to map the chemical composition of sample 112 with
nanometer-scale resolution. Note that, in order to enhance the
sensitivity of the measurement technique, the timing of the
pulse(s) may be matched with the duration of the probe tip-sample
contact during Z-modulation. The temporal width and/or intensity of
a pulse can be adjusted to maximize the contrast. By detecting and
recording changes in the parasitic-suppressed torsional deflections
in response to the applied pulse(s) and scanning cantilever 114
across the surface of sample 112, it may be possible to map the
chemical composition of sample 112.
[0073] In some embodiments, measurement circuit 124 measures a
vertical signal associated with relative displacement, along
direction 122, of sample 112 and probe tip 116. This vertical
signal is sometimes referred to as a `flexural signal` or a
`flexural deflection signal.` Moreover, AFM 100 may also further
determine the information based on the vertical signal.
[0074] For example, AFM 100 may generate at least one
force-distance curve and, more generally, at least one
force-distance curve at each pixel location of an image. The force
values may be determined from the parasitic-suppressed torsional
deflection signals, and the distance values may be determined from
the displacement of actuator 118 used for the Z-modulation and the
flexural deflection signals. The force-distance curves can be used
to measure one or more materials properties. By scanning cantilever
114 across the surface of sample 112, these measured quantities can
be used to generate images that map composition of materials, which
is particularly useful in characterization of heterogeneous
materials. Note that the vertical signals may be measured and used
in conjunction with the lateral signals to adjust the distance
values because of the position of probe tip 116. In particular,
this is because the actual position of probe tip 116 relative to
the surface is the sum of tip displacements due to flexural and
torsional motions, as well as the displacement of actuator 118 used
for Z-modulation. Moreover, note that the measurement technique may
reduce the uncertainties introduced by the recovery step or
operation that is often used to remove the parasitic signals.
[0075] In addition to correcting the lateral signals for the
baseline lateral signal value, AFM 100 may perform an adjustment to
minimize the effects of angular misalignment between the
orientation of cantilever 114 and the orientation of the
photo-detector in measurement circuit 124. If this angular
misalignment is large, it can exacerbate crosstalk from flexural
deflections into the lateral signals. Therefore, AFM 100 may
perform an adjustment to minimize the angular misalignment, such as
by adjusting the photo-detector orientation. Alternatively or
additionally, AFM 100 may determine the amount of crosstalk (such
as by comparing the vertical signal and the lateral signal from
cantilever 114 when it is not interacting with sample 112), and
then correcting for the crosstalk.
[0076] While the preceding embodiment included the use of feedback
(via optional feedback circuit 126, actuator 118 and/or optional
actuator 128), in other embodiments the information is determined
without the use of feedback. In particular, measurement circuit 124
may measure the lateral signal and the vertical signal. Then, AFM
100 may determine the information about sample 112 based on the
lateral signal and the vertical signal. Moreover, the determination
may involve correcting for parasitic signals in the lateral signal
and the vertical signal. The parasitic signals may correspond to
phenomena other than probe tip-sample interaction, thermal noise of
cantilever 114 and measurement-circuit noise.
[0077] Although we describe AFM 100 as an example, in alternative
embodiments, different numbers or types of components may be
present. For example, some embodiments comprise more or fewer
components. Alternatively or additionally, two or more components
may be combined together. Therefore, in some embodiments actuator
118 and optional actuator 128 are combined into a single actuator.
However, in other embodiments, actuator 118 and/or optional
actuator 128 are separate actuators. There may also be additional
actuators (not shown) that are more efficient at higher
frequencies.
[0078] We now further describe the measurement technique.
Peak-force-based AFMs often provide improved control of the imaging
process. In particular, instead of relying on the vibration
amplitude (as is usually the case in tapping-mode AFMs), the
peak-force value can offer a robust technique for detecting probe
tip-sample contact. However, parasitic deflection signals can
degrade the performance of peak-force-based AFMs by making it
difficult to accurately determine probe tip-sample forces.
[0079] FIG. 2 presents a drawing illustrating an example of
Z-modulation, a vertical signal and a probe tip-sample force
waveform as a function of time for a peak-force tapping AFM
operating in liquid. As shown in FIG. 2, the Z-modulation signal is
often sinusoidal with a fundamental frequency and amplitude.
[0080] In general, the probe tip-sample force waveform is usually
not directly detectible from the measured vertical deflections. In
particular, the probe tip-sample force waveform typically exhibits
an alternating pattern of attractive and repulsive forces varying
around a baseline value. Moreover, the vertical signal may not
follow a clear pattern that can directly reveal the probe
tip-sample forces. Instead, there is usually a background signal in
addition to the deflection signals that are generated in response
to the probe tip-sample forces. In FIG. 2, arrows indicate regions
210 where the probe tip-sample interaction occurs. In this example,
the value of the deflection signal in one of these regions is below
the peak vertical signal. Therefore, unless probe tip-sample forces
are high enough to cause deflection above the background, it may
not be possible to directly use the peak vertical signal for
feedback control.
[0081] Note that the background signals may be parasitic signals
that originate from the operation of the AFM (such as from sources
other than probe tip-sample interactions). For example, the
parasitic signals can include cantilever bending due to viscous
drag forces, acceleration due to Z-modulation, and/or laser
interference. The parasitic signals are usually oscillatory
signals, primarily at the fundamental frequency of the
Z-modulation. Moreover, parasitic signals due to laser interference
may exhibit frequency doubling. (In this definition, noise from the
thermally induced vibrations of the cantilever and photo-detector
noise may not be parasitic signals). In addition to the parasitic
signals, there may be additional signals due to the excitation of
cantilever resonances in response to abruptly changing adhesion
forces. These parasitic signals, which are often encountered when
imaging takes place in air due to strong capillary forces, are
sometimes referred to as' unwanted signals.' This is because the
unwanted signals distort the cantilever deflection signals and can
make it difficult to relate deflection signals to probe tip-sample
forces.
[0082] As noted previously, in order to obtain deflection signals
that are free from parasitic signals, existing peak-force tapping
AFMs typically employ a recovery step or operation. This recovery
step may be performed by a digital controller that includes an
analog-to-digital converter, a field programmable gate array
(FPGA), and/or a digital signal processor. For example, a
background signal associated with AFM operation may be determined
by lifting the probe tip up from the surface. Once the background
signal is determined, a background generator may synthesize a
correction signal that is then subtracted from the detected
deflection signal. In addition, existing AFMs may use another
operation to determine baseline signals associated with drifts in
cantilever deflection and laser position on the photo-detector.
This baseline force may be treated separately from parasitic
signals. Note that this other operation may be performed so that
the zero deflection signal of the baseline-corrected signal
corresponds to zero probe tip-sample force.
[0083] In contrast, the measurement technique may suppress
parasitic signals (e.g., by more than an order of magnitude) so
that probe tip-sample forces can be detected more accurately for
use in peak force feedback and to facilitate more accurate
measurements of materials properties. In particular, the AFM
cantilever may respond to vertical probe tip-sample forces by
torsional bending (i.e., twisting). Then, during the Z-modulation,
torsional deflections of the cantilever may result in deflection
signals in which parasitic signals are suppressed relative to the
signals that are in response to the probe tip-sample forces.
[0084] FIGS. 3A and 3B presents drawings illustrating an example of
a T-shaped cantilever 300 for use with AFM 100 (FIG. 1) that
strongly respond to vertical probe tip-sample forces by torsional
bending. In particular, FIG. 3A illustrates a torsional mode of
cantilever 300 and FIG. 3B illustrates a flexural mode of
cantilever 300.
[0085] FIG. 4 presents a drawing illustrating an example of
vertical signal and parasitic-suppressed lateral signal
measurements provided by a quadrant photo-detector in
peak-force-based AFM 100 (FIG. 1) with a T-shaped cantilever.
During Z-modulation, the relative distance between the cantilever
and the sample may be modulated using an actuator, so that the
probe tip approaches, interacts, and moves away from the sample
surface. Deflections of the cantilever in flexural and torsional
modes may be detected by the position-sensitive quadrant
photo-detector. In particular, the lateral signal from the
photo-detector may correspond to the torsional bending of the
cantilever, and the vertical signal may correspond to the flexural
bending of the cantilever. Moreover, the laterally offset position
of the probe tip may cause torsional bending in response to
vertical probe tip-sample forces. Furthermore, the cantilever may
respond to the probe tip-sample forces by bending in the flexural
mode. Therefore, both the vertical and lateral signals may convey
or include information about the probe tip-sample forces. However,
parasitic signals may be suppressed in the lateral signals. In
contrast with the vertical signals, the lateral signals may
facilitate the detection of peak forces, as well as in detecting
the entire probe tip-sample force waveform.
[0086] FIG. 5A presents a drawing illustrating an example of a
vertical signal 510 as a function of time. This vertical signal may
correspond to flexural bending of a T-shaped cantilever (with the
probe tip laterally offset from the longitudinal axis of torsion)
in a peak-force-based AFM while interacting with a sample.
Moreover, FIG. 5B presents a drawing illustrating an example of a
lateral signal 512 as a function of time. This lateral signal may
correspond to torsional bending of the T-shaped cantilever in the
peak-force-based AFM while interacting with a sample. In FIGS. 5A
and 5B, parasitic signals 514 and 516 are shown as dashed lines.
Note that parasitic signals 514 and 516 may be estimated by fitting
to a sinusoidal waveform. (Alternatively, parasitic signals 514 may
be estimated by lifting the cantilever away from the surface using
the recovery step or operation.) In some embodiments, the fitting
process excludes the contact zone, which is approximately 40% of
the Z-modulation period in this example.
[0087] Note that peak-to-peak parasitic deflection 518 is
substantially larger than vertical signal 510 corresponding to peak
force 520. In contrast, peak-to-peak parasitic deflection 522 is
substantially less than lateral signal 512 due to peak probe
tip-sample force 524. Consequently, by using lateral signal 512
instead of vertical signal 510, parasitic signal 516 is
substantially suppressed relative to lateral signal 512 in response
to peak probe tip-sample force 524. Therefore, in the measurement
technique lower probe tip-sample forces can be detected without
using a recovery step or operation. Stated differently, while the
same probe tip-sample force generates both vertical and lateral
signals 510 and 512, lateral signal 512 rises well above parasitic
signal 516 in the lateral channel and vertical signal 510 remains
below parasitic signal 514 in the vertical channel. Thus, imaging
can be performed with lower probe tip-sample forces without the
need for a recovery step or operation.
[0088] FIG. 6 presents a drawing illustrating an example of a
cantilever 600 for use with AFM 100 (FIG. 1). This cantilever
includes a cantilever body 610 having: a length 612, a stem or
lateral width 614, an arm length 616, an arm width 618, a reflector
width 620 and a reflector length 622. Moreover, cantilever 600 may
include a probe tip 624 that is offset 626 along a lateral
direction 628 from a longitudinal axis of torsion 630 of cantilever
600. Note that cantilever 600 may have a torsional mode that
suppresses parasitic signals, where the parasitic signals
correspond to phenomena other than probe tip-sample interaction,
thermal noise of the cantilever and measurement-circuit noise.
Therefore, cantilever 600 may be used to measure
parasitic-suppressed torsional deflection signals.
[0089] The geometry of cantilever 600 may reduce the parasitic
signals and force noise in peak-force-based AFMs without
significantly compromising noise performance (e.g., without
increasing thermal noise). For example, as a non-limiting example,
length 612 may be 115 .mu.m, lateral width 614 may be 9 .mu.m, arm
length 616 may be 60 .mu.m, arm width 618 may be 9 .mu.m, reflector
width 620 may be 20 .mu.m, reflector length 622 may be 25 .mu.m and
offset 626 may be greater than 2 .mu.m (such as 20-30 .mu.m, e.g.,
27 .mu.m). Note that offset 626 is intentional and, therefore, is
larger than accidental offsets than can occur due to fabrication
errors such as mask misalignment. Such unintentional offsets are
typically less than 2 .mu.m.
[0090] In some embodiments, a ratio of offset 626 of probe tip 624
along lateral direction 628 to a length 612 of cantilever body 610
(which is sometimes referred to as an `aspect ratio`) is greater
than or equal to 0.235 or is greater than or equal to 0.4. The
aspect ratio may impact the ratio of the torsional spring constant
to the flexural spring constant. Note that the spring constants in
the torsional and flexural modes each refer to ratio of the force
acting on probe tip 624 to the displacement of probe tip 624 in the
respective mode. (In general, probe-tip displacement is a
superposition of displacements in several flexural and torsional
modes, with the dominant ones being the modes with the lowest
flexural and torsional resonance frequencies, respectively.) In
general, the larger the aspect ratio, the lower the torsional
spring constant relative to the flexural spring constant.
Therefore, cantilever 600 may have a larger aspect ratio to achieve
lower torsional spring constants. In some embodiments, the
torsional spring constant is decreased by reducing a thickness of
cantilever 600. However, this change may also reduce the flexural
spring constant, which may make it difficult for probe tip 624 to
detach from the sample surface as the cantilever moves away from
the sample. In addition, a lower flexural spring constant can cause
a large parasitic signal in the vertical channel. Consequently, the
aspect ratio may be used to reduce the torsional spring constant
without compromising the performance of cantilever 600 with respect
to the suppression of the parasitic signals.
[0091] Moreover, a ratio of offset 626 to lateral width 614 of
cantilever body 610 (which is sometimes referred to as `an
offset-to-stem ratio`) may be greater than or equal to 3. This
geometry of cantilever 600 may offer reduced surface area for the
same lateral offset 626 and length 612). A reduced surface area may
result in lower fluid drag forces, while maintaining a high aspect
ratio to keep the torsional spring constant low relative to the
spring constant of the fundamental flexural mode. Lower fluid drag
forces may reduce: parasitic deflections in the flexural mode;
crosstalk from the vertical signal to the lateral signal; and/or
the thermal-noise-limited minimum detectible force. These
capabilities may allow peak forces to be detected accurately. In
this regard, arm width 618, reflector width 620 and reflector
length 622 may also be small. In general, the reflector area may be
sufficiently large to accommodate a laser spot of the AFM If the
AFM has a sufficiently small laser spot size, the reflector area
may be as narrow as lateral width 614. In embodiments where the
cantilever edges are not straight lines (so that the lateral width
cannot be determined clearly), the average lateral width (i.e., the
average width in lateral direction 628) can be used. Because the
narrowest regions dominate the torsional spring constant, the
averaging may be performed over the regions along length 612 of
cantilever 600 whose widths belong to the lowest 50% of the widths
along length 612. If there are multiple stems that are connected to
the arm that holds probe tip 624 or the reflector area, the width
may be calculated as the total width in lateral direction 628
(i.e., the sum over the stems).
[0092] Furthermore, the torsional spring constant of the
fundamental flexural modes of cantilever 600, which may be defined
as the ratio of the force acting on probe tip 624 and along a
direction that is perpendicular to the cantilever surface (such as
the surface that reflects the laser light used for deflection
detection) to the probe-tip displacement caused by torsional
deflections, may be between 0.01 and 10,000 N/m. As noted
previously, if cantilever 600 has a spring constant that is too low
(lower than about 0.01 N/m), it may be difficult to break probe
tip-sample contact due to adhesive forces. In addition, if
cantilever 600 has spring constant that is too high (higher than
about 1,000 N/m), it may be difficult to maintain a low enough peak
force during the imaging process. The wide range of spring
constants of cantilever 600 may provide the capability to measure
and map mechanical properties over a wide range, from 100 Pa to 100
GPa. The lower range may be smaller than the value in existing
peak-force tapping AFMs that rely on a recovery step or operation
to obtain deflection signals. Thus, cantilever 600 may make it
possible to detect probe tip-sample force waveforms on more
compliant samples, which may be difficult to measure accurately due
to difficulties in the recovery step or operation.
[0093] Additionally, cantilever body 610 may include silicon and/or
silicon nitride, and probe tip 624 may include silicon, silicon
dioxide and diamond. Cantilever 600 may extend from a support
structure (not shown) at base 632 of cantilever body 610. Moreover,
as a non-limiting example, cantilever body 610 may have a thickness
of approximately 600 nm. Furthermore, there may be a gold coating
having an approximate thickness of 30 nm on a back surface of
cantilever body 610 (facing away from probe tip 624) to enhance
reflectivity of the laser beam used in deflection detection.
Cantilever 600 may be manufactured using a variety of lithographic
fabrication techniques, including additive and subtractive
processes. Note that probe tip 624 may be chemically functionalized
to alter probe tip-sample interaction forces, e.g., to reduce or
enhance capillary forces and/or electrical forces.
[0094] As shown in FIG. 6, cantilever 600 may have a T-shape (in
addition to the lateral displacement of probe tip 624). However, in
other embodiments another geometry may be used to provide
parasitic-suppressed torsional deflection signals. In particular,
cantilever 600 may have a geometry in which probe tip-sample
interaction forces generate sufficient torsional deflection signals
to exceed the parasitic signals. In general, the suitability of a
particular cantilever geometry for suppressing parasitic signals
can be tested in a peak-force tapping AFM (e.g. using vertical
signals to determine peak forces for feedback control) and
comparing the lateral and vertical signals. If the magnitude of the
parasitic signal is smaller than the magnitude of the lateral
signal associated with the peak probe tip-sample force (compared to
the relative magnitude of the vertical signal), then this
cantilever geometry may provide parasitic-suppressed torsional
deflection signals. Note that such cantilevers are sometimes
referred as `parasitic-suppressed torsional cantilevers.`
[0095] Note that longitudinal axis of torsion 630 may refer to an
axis of the cantilever 600 where cantilever 600 is not displaced by
torsional vibration when cantilever 600 is vibrate, i.e., the axis
of torsion is where cantilever 600 does not move in the torsional
mode. More specifically, the axis of torsion generally extends in
longitudinal direction 634 of cantilever 600. For a symmetrical
cantilever, such as rectangular cantilever 600, the axis of torsion
is the centerline of cantilever body 610 perpendicular to base 632
of cantilever 600. However, for a cantilever having a different
geometry, the axis of torsion may not necessarily be the centerline
of the cantilever body and/or may not be a straight line. Moreover,
depending of the placement of probe tip 624, deflection signals
corresponding to a positive probe tip-sample force (pointing away
from the sample surface towards the probe) can have a positive or
negative value. For example, probe tip 624 may be placed on the
left side of longitudinal axis of torsion 630 instead of the right
side (as shown in FIG. 6), which would cause cantilever 600 to
twist in the opposite direction in response to probe tip-sample
forces. In the present disclosure, the convention is that positive
(repulsive) probe tip-sample forces cause positive torsional
deflection signals.
[0096] As discussed previously, in addition to using a
parasitic-suppressed torsional cantilever to image a sample
(including varying the distance between the cantilever and the
sample surface so that the probe tip approaches, interacts and
moves away from the surface, and measuring vertical and/or lateral
signals in which the parasitic signals are suppressed by the
torsional mode of the cantilever), the measurement technique may
include a residual recovery operation to estimate and remove
residual parasitic signals in the parasitic-suppressed torsional
deflection signals.
[0097] This is shown in FIG. 7, which presents a block diagram
illustrating an example of a parasitic lateral-signal estimation
circuit 700 for use with AFM 100 (FIG. 1). In particular, parasitic
lateral signal estimator 710 may use parasitic-suppressed torsional
deflection signal 712 to estimate residual parasitic signal 714 in
the lateral signals. The estimation operation may include
separating the probe tip from the sample while feedback is turned
off, measuring the deflection signals in the separated
configuration, synthesizing a signal that replicates the measured
signals (i.e., residual parasitic signal 714), subtracting the
synthesized signal from the detected signals, and turning he
feedback on to bring the probe tip back into contact with the
sample. In parasitic lateral-signal estimation circuit 700,
subtraction circuit 716 may subtract residual parasitic signal 714
from parasitic-suppressed torsional deflection signal 712 to obtain
a lateral signal 718 that has lower contribution from parasitic
signals, resulting in further improvements in the detection of
probe tip-sample forces. The resulting torsional deflection or
lateral signal 718 may be used for feedback in a peak-force-based
AFM.
[0098] Note that the removal of residual parasitic signal 714 in
the parasitic-suppressed torsional deflection signal 712 (the
aforementioned residual recovery operation) may not be needed for
peak-force-based feedback control when parasitic-suppressed
torsional signal 712 are used to determine the peak forces. This is
because the parasitic signals may already be suppressed
substantially, and therefore very small probe tip-sample forces are
sufficient to overcome feedback errors introduced by the remaining
parasitic signals. However, removal of residual parasitic signals
can further improve force-distance curve measurements, which may be
used to measure materials properties, such as: adhesion,
elasticity, Young's modulus, and/or dissipation.
[0099] In some embodiments, an analog approach is used to reduce
crosstalk in the measurement technique. In particular, residual
parasitic signal 714 in parasitic-suppressed torsional deflection
signal 712 can include a crosstalk signal from the vertical signal
into the lateral signal. While the residual recovery operation can
be used to estimate and remove a residual parasitic signal
associated with the crosstalk, separately or additionally the
crosstalk may be eliminated using a calibration operation. For
example, the crosstalk may be minimized by adjusting the relative
orientation of the quadrant photo-detector. In particular, after
applying a Z-modulation signal (while the probe tip is separated
from the sample), the orientation of the photo-detector may be
adjusted to minimize the magnitude of the lateral signal.
(Alternatively, this adjustment may be performed electronically.)
Because the relative orientation of the cantilever can affect the
degree of crosstalk, this calibration operation may be performed
after placing a new cantilever in an AFM.
[0100] Alternatively or additionally, in some embodiments a digital
approach is used to reduce crosstalk in the measurement technique.
In particular, while the probe tip is separated from the sample, a
Z-modulation signal can be applied, and vertical and lateral
detector signals can be analyzed digitally to determine the
crosstalk ratio. The crosstalk ratio may be determined by
subtracting the respective baseline deflections from each of the
vertical and lateral signals. Then, the scalar ratio can be
determined based on a linear fit between the resulting lateral
signal and the resulting vertical signal (e.g., by using a
least-squares fitting technique). Once the crosstalk ratio is
determined, the lateral signal may be redefined to account for the
crosstalk. For example, the instantaneous vertical signal may be
divided by the crosstalk ratio, and then the result may be
subtracted from the raw lateral signal to obtain a redefined
lateral signal. The redefined lateral signal may also be a
parasitic-suppressed torsional deflection signal and its parasitic
component may have a lower contribution from crosstalk. Note that
the Z-modulation signal may have the fundamental frequency that
will be used during the measurement and imaging processes, because
the magnitude of the crosstalk may depend on the modulation
frequency (e.g., different flexural modes can have different
crosstalk ratios and the fundamental frequency of the Z-modulation
can affect the relative contributions from each flexural mode to
the overall cantilever motion). Therefore, the degree of crosstalk
can be minimized while engaging the probe tip to the sample. In
some embodiments, the crosstalk ratio is calculated using a
field-programmable gate array (FPGA), and the redefined lateral
signal is determined from the raw vertical and lateral detector
signals using an FPGA. Unlike the residual recovery operation, the
calibration operation may reduce the crosstalk independent of the
Z-modulation amplitude, because the crosstalk ratio may be
substantially independent of the Z-modulation amplitude.
[0101] Another approach for estimating the crosstalk ratio during
the imaging process involves determining the ratio of the
baseline-corrected vertical signal to the baseline-corrected
parasitic-suppressed torsional deflection signal using signals that
are outside of predetermined intervals (which are sometimes
referred to as `interaction windows`) that approximate the duration
of the probe tip-sample interaction during Z-modulation. This is
shown in FIG. 8, which presents a drawing illustrating an example
of Z-modulation 810, a vertical signal 812 and a
parasitic-suppressed lateral signal 814 as a function of time. The
predetermined intervals 816 may be chosen to be 40% of the period
of Z-modulation 810 and they can be centered on a time that
corresponds to the peak force value (prior to the subtraction of
the residual forces). In compliant regions of samples, the
predetermined interval can be chosen to be longer to ensure that
the probe tip-sample forces are excluded from the calculation of
the crosstalk ratio. Moreover, the crosstalk ratio may be estimated
at the beginning of a scanning process during imaging of a sample,
and it can be updated using measurements of the vertical and
lateral signals during the imaging process. Once the crosstalk
ratio is determined, the baseline-corrected vertical signals can be
divided by the crosstalk ratio and the result can be subtracted
from the baseline-corrected parasitic-suppressed torsional
deflection signals to obtain the redefined lateral signals.
[0102] In some embodiments, the AFM estimates and subtracts the
residual parasitic signals and/or determines the crosstalk ratio
and the redefined lateral signals using one or more
analog-to-digital (A/D) converters, one or more digital signal
processors, and/or one or more FPGAs. By performing the
calculations digitally, rapid feedback may be provided based on the
peak deflection or force value.
[0103] Note that peak-force-based AFMs typically offer improved
control and faster feedback relative to tapping-mode AFMs. In
contrast with peak-force-based AFMs, tapping-mode AFMs usually rely
on changes in the vibration amplitude for feedback, which often
exhibit complicated dynamics and slow transients. In
peak-force-based AFMs, as soon as the value of peak force is
determined, the feedback signal can be adjusted before the
oscillation cycle is completed. The feedback loop may use an
actuator (such as a piezoelectric actuator) to adjust the relative
position of the cantilever and the sample so that the peak force is
restored to its set point value in the subsequent cycles of the
periodic Z-modulation. Because peak forces are typically
encountered when the tip is at its lowest point in its trajectory,
the measured value of the parasitic-suppressed torsional deflection
signal at this point can be used as the peak force signal.
Therefore, A/D converter(s), digital filtering, baseline
subtraction, and/or background generation and subtraction can be
used with the parasitic-suppressed torsional deflection signals
from an AFM operating in peak-force tapping-mode (i.e., by relying
on the lateral signal rather than the vertical signal to obtain
peak probe tip-sample forces). Note that because the parasitic
signals are greatly suppressed in the measurement technique, a
recovery step or operation may not be necessary. In addition,
additional techniques for determining the feedback control signal
(such as a predetermined synchronization distance and/or gated
averaging) may be used with the parasitic-suppressed torsional
deflection signals.
[0104] While the peak force value may be used in the feedback loop,
force values at any other time point within the Z-modulation cycle
can be used. In these embodiments, the center point of the gating
interval (i.e., the time window used to determine the peak force
value, such as by averaging forces during the gating interval) may
be adjusted to a desired time point. Furthermore, rather than
averaging the signals within the gating interval, it is also
possible to apply a weighted average. The use of gated averaging
may allow exclusion of time points at which the probe tip-sample
forces are not substantially larger than noise (e.g., thermal
noise) and/or the parasitic deflections. Moreover, the weighted
averaging can be used to give larger weights to the time points at
which the measured forces are larger than other time points within
the gating interval. Furthermore, weighted averaging of other
functions of the probe tip-sample force waveform (such as the
difference between the maximum and minimum force values) can also
be used for the feedback.
[0105] While force is the physical quantity representing the
interaction between the probe tip and the sample, for the purpose
of feedback deflection signals can also be used directly, without
determining a calibrated force value. For example, the feedback
loop can maintain a peak deflection signal during the imaging
process. Alternatively or additionally, parasitic-suppressed
torsional deflection signals at other time points within the
Z-modulation cycle or a weighted averaging of deflection signals
can also be used.
[0106] In some embodiments, the parasitic signals in the
parasitic-suppressed torsional signals result from crosstalk from
the vertical signals. These vertical signals may be due to
acceleration as a result of the Z-modulation, fluid-drag forces
and/or unwanted excitation of slowly decaying fundamental flexural
resonance. The parasitic signals may be reduced by only moving the
sample for Z-modulation. This is because, if the cantilever is
being moved for Z-modulation (e.g., while keeping sample fixed in
the Z direction perpendicular or approximately perpendicular to the
sample surface), the actuator may excite resonances if the
fundamental frequency is at or near the resonance frequency of a
flexural mode. However, many AFMs are equipped with actuators that
move the cantilever in the Z direction. This configuration may
allow simultaneous imaging using optical microscopy and AFM. In
order to further suppress the parasitic signals associated with
crosstalk into torsional deflection signals, in an AFM that moves
the cantilever for Z-modulation, the modulation fundamental
frequency may be less than the fundamental flexural resonance
frequency of the cantilever. Alternatively, resonance effects may
dominate the probe-tip displacement (and, therefore, the parasitic
flexural deflection signals) when the Z-modulation fundamental
frequency is within the resonance peak defined by the quality
factor of the cantilever. Therefore, the Z-modulation may be kept
below the resonance peak. (Choosing a modulation fundamental
frequency above the resonance peak may result in the cantilever
moving in the opposite direction or out of phase, which may enhance
the flexural deflection signal, and therefore the parasitic signals
in the lateral deflections due to crosstalk.) Note that the
boundaries of the resonance peak can be determined from
measurements of the fundamental flexural resonance frequency and
its quality factor (such as using the thermal noise spectrum). The
frequency at which the cantilever response (such as the amplitude
of the thermal noise, or the vibration amplitude if frequency
tuning is used) is half of its peak value at the resonance
frequency may be the lower boundary of the resonance peak. If the
resonance frequency and quality factor of the cantilever are known,
the lower bound of the resonance peak can be calculated according
to:
f low = f res - f res 2 Q , ##EQU00001##
where f.sub.low is the lower bound of the resonance peak, f.sub.res
is the flexural resonance frequency of the cantilever measured in
the imaging fluid, and Q is the quality factor of the fundamental
flexural mode. (Note that the parameters may be measured in the
imaging medium and in the vicinity of the sample to account for
squeezed-film damping effects.) For example, if the resonance
frequency of the cantilever is 50 kHz and its Q is 25, then the
lower bound is 49 kHz. In this example, the drive or modulation
fundamental frequency may be below 49 kHz.
[0107] In embodiments that move the sample during the Z-modulation,
there may be a benefit in using a Z-modulation fundamental
frequency that is below the resonance peak because viscous drag
forces can mechanically couple the sample to the cantilever. This
effect may be weaker than the effect in embodiments in which the
cantilever is moved during the Z-modulation, because
acceleration-related excitation of cantilever movements may be
prevented.
[0108] Although choosing a Z-modulation fundamental frequency below
the resonance peak may reduce the crosstalk signals associated with
the parasitic excitation of a flexural resonance, choosing an even
lower Z-modulation fundamental frequency may keep
acceleration-related parasitic flexural deflection of the
cantilever small. This is illustrated in FIG. 9, which presents a
drawing of an example of a model 900 of parasitic flexural
deflections during Z-modulation of a cantilever. In particular, the
fundamental flexural mode of the cantilever can be approximated by
a damped simple harmonic oscillator (higher-order modes typically
have high resonance frequencies and spring constants, and their
effects in this analysis are typically negligible). The actuator
used for the Z-modulation may move the cantilever base at an
angular frequency w. However, while the cantilever body may
accelerate according to the base displacement, the probe-tip
trajectory (the position versus time curve) may not be identical to
the base displacement. In the frequency domain, the equation of
motion of the probe-tip mass (e.g., the equation of motion
corresponding to the equivalent mass in the simple harmonic
oscillator model), which relates the displacement of the probe tip
to the displacement of the cantilever base, may be expressed
as:
X = A ( 1 - w 2 w o 2 ) - j w Q w o , ##EQU00002##
where X and A are the frequency-dependent complex (having real and
imaginary parts) values corresponding to the probe-tip displacement
and cantilever-base displacement (i.e., X and A are the Fourier
transforms of the time-dependent displacements), w is the
fundamental frequency of Z-modulation, w.sub.0 and Q are the
resonance frequency and quality factor of the fundamental flexural
mode in the medium of imaging, and j is the imaginary unit (i.e.,
the square root of -1). In order to ensure that the cantilever-base
displacement dominates the probe-tip displacement (i.e., the
parasitic flexural deflections induced by the Z-modulation are less
than Z-modulation distance of the base), X and A may be constrained
according to
|X-A|<|A|.
In this inequality, the left side is the magnitude of the parasitic
flexural deflections and the right side is the magnitude of the
cantilever-base modulation distance. Therefore, in order to
dominate the probe-tip displacement, the cantilever-base
displacement may have to have a larger magnitude than the parasitic
flexural deflections. Using the frequency-dependent relationship
between X and A, it can be shown that, in order for the above
inequality to hold, the Z-modulation fundamental frequency may be
less than a threshold frequency, f.sub.threshold, i.e.,
f threshold = f res 2 . ##EQU00003##
Consequently, in order to ensure that the Z-modulation distance at
the cantilever base dominates the probe-tip displacement, the
modulation fundamental frequency may be below the threshold
frequency (approximately 0.707 times the resonance frequency of the
fundamental flexural mode in the imaging medium).
[0109] Note that we defined two frequency values (f.sub.low and
f.sub.threshold) to ensure that parasitic signals in the torsional
deflection signals are not increased by parasitic flexural
deflections via crosstalk of signals. Depending on the quality
factor of the fundamental flexural resonance frequency, one or the
other frequency value may be smaller. The lower of the two
frequency values, which may be defined as the critical frequency
f.sub.critical, may be used in the measurement technique.
[0110] In embodiments that use fundamental frequencies at or near a
flexural resonance frequency of the cantilever, an analog or a
digital technique for crosstalk elimination may be used. By
reducing the parasitic signals in the lateral signal, these
embodiments may allow the use of fundamental frequencies such as
those that are used in the tapping mode, while using the
parasitic-suppressed torsional deflection signals (i.e., the
lateral signal) to obtain the feedback signal. In these
embodiments, the feedback signal may be based on a peak force, an
average force during a gating interval, and/or a weighted average
force during a gating interval. Furthermore, these forces may be
synchronously averaged over many cycles of the fundamental
frequency of the tip oscillation. Alternatively, the peak forces,
average forces during the gating interval, and/or the weighted
average force may be determined from a synchronously averaged
tip-sample force waveform at the fundamental frequency of the tip
oscillation. By relying on peak forces during the imaging process,
this approach may offer faster feedback and more robust operation,
in comparison to existing tapping-mode AFM measurement
techniques.
[0111] We now describe embodiments of a method in the measurement
technique. FIG. 10 presents a flow diagram illustrating an example
of a method 1000 for determining information about a sample based
on a lateral signal using an AFM, such as AFM 100 (FIG. 1). During
operation, the AFM may vary a distance between the sample and a
probe tip (operation 1010) along a direction approximately
perpendicular to a plane of the sample in an intermittent contact
mode, where the probe tip is included in a cantilever and is offset
along a lateral direction from a longitudinal axis of torsion of
the cantilever. Then, the AFM may measure the lateral signal
(operation 1012) associated with a torsional mode of the cantilever
during AFM measurements, where the lateral signal corresponds to a
force between the sample and the probe tip;
[0112] Moreover, the AFM may maintain, using a feedback circuit in
the AFM and relative to a threshold value, a parameter (operation
1014), such as: the force between the sample and the probe tip,
and/or a deflection of the cantilever corresponding to the force.
Note that maintaining the force may involve changing the distance
between the sample and the probe tip along the direction.
[0113] Next, the AFM may determine the information about the sample
(operation 1016) based on at least the lateral signal.
[0114] As noted previously, in some embodiments the measurement
technique is used without feedback. This is shown in FIG. 11, which
presents a flow diagram illustrating an example of a method 1100
for determining information about a sample using an AFM, such as
AFM 100 (FIG. 1). During operation, the AFM may vary a distance
between the sample and a probe tip (operation 1110) along a
direction approximately perpendicular to a plane of the sample in
an intermittent contact mode, where the probe tip is included in a
cantilever and is offset along a lateral direction from a
longitudinal axis of torsion of the cantilever.
[0115] Then, the AFM may measure a lateral signal (operation 1112)
associated with a torsional mode of the cantilever during AFM
measurements and may measure a vertical signal (operation 1114)
associated with relative displacement, along the direction, of the
probe tip and the sample, where the lateral signal corresponding to
a force between the sample and the probe tip.
[0116] Next, the AFM may determine the information about the sample
(operation 1116) based on the lateral signal and the vertical
signal. For example, the information may include material
properties derived from force curves obtained using the lateral
signal and the vertical signal during Z-modulation (such as from
force-distance curves). In some embodiments, the information may
include a mapping of other aspects of the probe tip-sample
interaction, such as: the indentation distance, energy dissipation
(the area inside the force curve), and/or electrical properties if
a voltage signal is applied to the probe tip relative to the
sample.
[0117] In some embodiments, an electronic device (which is
sometimes referred to as an `instrument module`) is used in
conjunction with an AFM (which may be an existing AFM) and a
parasitic-suppressed torsional cantilever to perform the
measurement technique. FIG. 12 presents a block diagram
illustrating an example of an electronic device 1200 for use with
an AFM, such as AFM 100 (FIG. 1). This electronic device may
include input nodes 1210 that couple to a measurement circuit in
the AFM and that receive, from the measurement circuit, a
measurement signal, where the measurement signal includes a lateral
signal associated with a torsional mode of a cantilever in the AFM
during AFM measurements, and the lateral signal corresponds to a
force between a sample and a probe tip in the cantilever. Moreover,
electronic device 1200 may include input nodes 1212 that couple to
a feedback circuit in the AFM and that receive, from the feedback
circuit, a feedback signal, where the feedback signal corresponds
to a vertical signal associated with relative displacement, along a
direction approximately perpendicular to a plane of the sample, of
the probe tip and the sample. Furthermore, electronic device 1200
may include a signal-conditioning circuit 1214 that modifies the
feedback signal so that the modified signal corresponds to a force
between the sample and the probe tip. Additionally, electronic
device 1200 may include output nodes 1216 that couple to the
feedback circuit and that provide the measurement signal to the
feedback circuit, and output nodes 1218 that couple to the
measurement circuit and that provide the modified feedback signal
to the measurement circuit. Note that signal-condition circuit 1214
may apply a feed-forward modification to the feedback signal,
which, in part, may be based on a transfer function and/or desired
signal conditioning. Thus, electronic device 1200 may provide a
deliberately different or modified feedback signal to the AFM This
may overcome a speed limitation of the AFM
[0118] Thus, electronic device 1200 may facilitate process the
parasitic-suppressed torsional deflection signal (i.e., the lateral
signal) and to provide the AFM with one or more signals including,
but not limited to: a feedback error signal, values of materials
properties, and/or a waveform derived from the parasitic-suppressed
torsional deflection (such as a derived deflection waveform).
[0119] In some embodiments, electronic device 1200 receives, from
the AFM, one or more additional signals, including: a vertical
signal, a Z-modulation signal, signals containing information about
the relative position of the sample and the probe tip, an optional
electrical bias applied to the probe tip and/or the sample, and/or
a trigger signal for electromagnetic and optical pulses applied to
the tip and/or the sample. In general, the input signal to and
output signals from electronic device 1200 can be analog and/or
digital signals.
[0120] Note that electronic device 1200 may include one or more
processors or micro-controllers, one or more FPGAs, and/or one or
more A/D converters that can sample the vertical and lateral
signals. The processor(s) and/or FPGAs can use the digitized
signals from the A/D converter(s) to process the
parasitic-suppressed torsional deflection signal. The processing
can include operations such as: determining a crosstalk ratio,
obtaining redefined lateral signals from the crosstalk ratio,
and/or determining and removing a residual parasitic signal.
Moreover, the processing may include filtering and scaling of
signals. Furthermore, electronic device 1200 may include one or
more digital-to-analog (D/A) converters so that calculated signals
can be converted to analog signals.
[0121] In some embodiments, electronic device 1200 is interfaced
with the AFM such that the feedback error signal calculated by
electronic device 1200 (e.g., based on the peak force or peak
deflection signal) can replace the original feedback error signal
of the AFM. Alternatively or additionally, electronic device 1200
may be interfaced with the AFM such that the waveform calculated by
electronic device 1200 (e.g., a derived deflection waveform) can be
directly used by the AFM feedback circuit or controller. In this
way, the AFM may process the calculated waveform signal to
determine its own feedback signal. Note that electronic device 1200
may include additional inputs and/or outputs dedicated to
communicate with a computer or workstation (which may be included
in or separate from the AFM) to transfer data and to adjust
settings of techniques used by the processor(s) and/or the
FPGAs.
[0122] The preceding apparatuses may include fewer or additional
components, the positions of one or more components may be moved
two or more components may be combined into a single component
and/or a single component may be separated into two or more
separate components. For example, electronic device 1200 may
include: scaling amplifiers, summing amplifiers, filters, and/or an
analog sample-and-hold circuit that samples the deflection signal
at a predetermined synchronization distance (such as the
synchronization distance that is set to the time when peak forces
are observed). The synchronization distance can be determined from
the Z-modulation signal input to electronic device 1200 and/or
using a peak-detection circuit that processes the
parasitic-suppressed torsional signal. Moreover, the output of the
sample-and-hold circuit can be provided as one of the outputs of
electronic device 1200 to the AFM feedback circuit or controller to
be used as the feedback signal. Furthermore, electronic device 1200
may output a signal waveform that is a linearly scaled version of
the parasitic-suppressed torsional deflection signal, and the AFM
may use this waveform to determine the feedback signal used for
tracking the surface topography. Additionally, electronic device
1200 may directly provide the parasitic-suppressed torsional
deflection signal to the AFM for use in determining the feedback
signal. In these embodiments, electronic device 1200 may
essentially swap the vertical signal with the parasitic-suppressed
torsional signal or the lateral signal.
[0123] FIG. 13 is a flow diagram illustrating an example of a
method 1300 for modifying a feedback signal using an electronic
device, such as electronic device 1200 (FIG. 12). During operation,
the electronic device may receive, on first input nodes from a
measurement circuit in an AFM, a measurement signal (operation
1310), where the measurement signal includes a lateral signal
associated with a torsional mode of a cantilever in the AFM during
AFM measurements, and the lateral signal corresponds to a force
between a sample and a probe tip in the cantilever. Moreover, the
electronic device may receive, on second input nodes from a
feedback circuit in the AFM, a feedback signal (operation 1312),
where the feedback signal corresponds to a vertical signal
associated with relative displacement, along a direction
approximately perpendicular to a plane of the sample, of the probe
tip and the sample.
[0124] Then, the electronic device may modify, using a
signal-conditioning circuit, the feedback signal (operation 1314)
so that the modified signal corresponds to a force between the
sample and the probe tip. For example, the signal-condition circuit
may apply a feed-forward modification to the feedback signal.
[0125] Furthermore, the electronic device may provide, on first
output nodes, the measurement signal (operation 1316) to the
feedback circuit. Additionally, the electronic device may provide,
on second output nodes, the modified feedback signal (operation
1318) to the measurement circuit.
[0126] In some embodiments of methods 1000, 1100 and/or 1300, there
may be additional or fewer operations. Moreover, the order of the
operations may be changed, and/or two or more operations may be
combined into a single operation. For example, in method 1300,
instead of receiving the feedback signal as an input, in some
embodiments the signal-conditioning circuit computes the feedback
signal based on the measurement signal and provides it to the
feedback circuit in the AFM.
[0127] We now describe embodiments of an electronic device. FIG. 14
presents a block diagram illustrating an example of an electronic
device 1400, such as AFM 100 in FIG. 1. This electronic device
includes processing subsystem 1410, memory subsystem 1412,
networking subsystem 1414, measurement subsystem 1426 and/or
optional feedback subsystem 1432. Processing subsystem 1410
includes one or more devices configured to perform computational
operations. For example, processing subsystem 1410 can include one
or more microprocessors, one or more GPUs, one or more
application-specific integrated circuits (ASICs), one or more
microcontrollers, one or more programmable-logic devices (such as
FPGAs), and/or one or more digital signal processors (DSPs).
[0128] Memory subsystem 1412 includes one or more devices for
storing data and/or instructions for processing subsystem 1410 and
networking subsystem 1414. For example, memory subsystem 1412 can
include dynamic random access memory (DRAM), static random access
memory (SRAM), and/or other types of memory. In some embodiments,
instructions for processing subsystem 1410 in memory subsystem 1412
include: one or more program modules or sets of instructions (such
as program module 1422 or operating system 1424), which may be
executed by processing subsystem 1410. Note that the one or more
computer programs may constitute a computer-program mechanism.
Moreover, instructions in the various modules in memory subsystem
1412 may be implemented in: a high-level procedural language, an
object-oriented programming language, and/or in an assembly or
machine language. Furthermore, the programming language may be
compiled or interpreted, e.g., configurable or configured (which
may be used interchangeably in this discussion), to be executed by
processing subsystem 1410.
[0129] In addition, memory subsystem 1412 can include mechanisms
for controlling access to the memory. In some embodiments, memory
subsystem 1412 includes a memory hierarchy that comprises one or
more caches coupled to a memory in electronic device 1400. In some
of these embodiments, one or more of the caches is located in
processing subsystem 1410.
[0130] In some embodiments, memory subsystem 1412 is coupled to one
or more high-capacity mass-storage devices (not shown). For
example, memory subsystem 1412 can be coupled to a magnetic or
optical drive, a solid-state drive, or another type of mass-storage
device. In these embodiments, memory subsystem 1412 can be used by
electronic device 1400 as fast-access storage for often-used data,
while the mass-storage device is used to store less frequently used
data.
[0131] Networking subsystem 1414 includes one or more devices
configured to couple to and communicate on a wired and/or wireless
network (i.e., to perform network operations), including: control
logic 1416, an interface circuit 1418, one or more antennas 1420
and/or input/output (I/O) port 1430. (While FIG. 14 includes one or
more antennas 1420, in some embodiments electronic device 1400
includes one or more nodes 1408, e.g., a pad, which can be coupled
to one or more antennas 1420. Thus, electronic device 1400 may or
may not include one or more antennas 1420.) For example, networking
subsystem 1414 can include a Bluetooth networking system, a
cellular networking system (e.g., a 3G/4G/5G network such as UMTS,
LTE, etc.), a universal serial bus (USB) networking system, a
networking system based on the standards described in IEEE 802.11
(e.g., a Wi-Fi networking system), an Ethernet networking system,
and/or another networking system.
[0132] Networking subsystem 1414 includes processors, controllers,
radios/antennas, sockets/plugs, and/or other devices used for
coupling to, communicating on, and handling data and events for
each supported networking system. Note that mechanisms used for
coupling to, communicating on, and handling data and events on the
network for each network system are sometimes collectively referred
to as a `network interface` for the network system. Moreover, in
some embodiments a `network` between the electronic devices does
not yet exist. Therefore, electronic device 1400 may use the
mechanisms in networking subsystem 1414 for performing simple
wireless communication between the electronic devices, e.g.,
transmitting advertising or beacon frames and/or scanning for
advertising frames transmitted by other electronic devices.
[0133] Measurement subsystem 1426 may include a
parasitic-suppressed torsional cantilever, a driver or an actuator,
a laser and/or quadrant photo-detector to perform the measurement
technique. Thus, measurement subsystem 1426 may determine the
lateral signal and/or the vertical signal. Moreover, optional
feedback subsystem 1432 may use at least the lateral signal in a
feedback-control loop.
[0134] In some embodiments, electronic device 1400 includes a
display subsystem (not shown) for displaying information on a
display, which may include a display driver and the display, such
as a liquid-crystal display, a multi-touch touchscreen, etc. For
example, the display may display an image acquired during a scan of
a sample.
[0135] Within electronic device 1400, processing subsystem 1410,
memory subsystem 1412, networking subsystem 1414, measurement
subsystem and/or optional feedback subsystem may be coupled
together using bus 1428. Bus 1428 may include an electrical,
optical, and/or electro-optical connection that the subsystems can
use to communicate commands and data among one another. Although
only one bus 1428 is shown for clarity, different embodiments can
include a different number or configuration of electrical, optical,
and/or electro-optical connections among the subsystems.
[0136] Electronic device 1400 can be (or can be included in) any
electronic device with at least one network interface. For example,
electronic device 1400 can be (or can be included in): a desktop
computer, a laptop computer, a subnotebook/netbook, a server, a
tablet computer, a smartphone, a cellular telephone, a smartwatch,
a consumer-electronic device, a portable computing device, an AFM,
another measurement device and/or another electronic device.
[0137] Although specific components are used to describe electronic
device 1400, in alternative embodiments, different components
and/or subsystems may be present in electronic device 1400. For
example, electronic device 1400 may include one or more additional
processing subsystems, memory subsystems, networking subsystems,
measurement subsystems, feedback subsystems, display subsystems
and/or signal-processing subsystems (such as A/D converters or D/A
converters). Additionally, one or more of the subsystems may not be
present in electronic device 1400. Moreover, in some embodiments,
electronic device 1400 may include one or more additional
subsystems that are not shown in FIG. 14. Also, although separate
subsystems are shown in FIG. 14, in some embodiments, some or all
of a given subsystem or component can be integrated into one or
more of the other subsystems or component(s) in electronic device
1400. For example, in some embodiments program module 1422 is
included in operating system 1424.
[0138] Moreover, the circuits and components in electronic device
1400 may be implemented using any combination of analog and/or
digital circuitry, including: bipolar, PMOS and/or NMOS gates or
transistors. Furthermore, signals in these embodiments may include
digital signals that have approximately discrete values and/or
analog signals that have continuous values. Additionally,
components and circuits may be single-ended or differential, and
power supplies may be unipolar or bipolar.
[0139] An integrated circuit may implement some or all of the
functionality of electronic device 1400. Moreover, the integrated
circuit may include hardware and/or software components that are
used for performing at least some of the operations in the
measurement technique.
[0140] While AFM is used as an illustrative example of the
measurement technique, the described embodiments of the measurement
technique may be used in a variety of measurement devices.
Furthermore, while some of the operations in the preceding
embodiments were implemented in hardware or software, in general
the operations in the preceding embodiments can be implemented in a
wide variety of configurations and architectures. Therefore, some
or all of the operations in the preceding embodiments may be
performed in hardware, in software or both. For example, at least
some of the operations in the measurement technique may be
implemented using program module 1422, operating system 1424, a
driver for interface circuit 1418 and/or in firmware in a hardware
component in electronic device 1400 (such as firmware in interface
circuit 1418). Alternatively or additionally, at least some of the
operations in the measurement technique may be implemented in a
physical layer, such as hardware in interface circuit 1418.
[0141] In the preceding description, we refer to `some
embodiments.` Note that `some embodiments` describes a subset of
all of the possible embodiments, but does not always specify the
same subset of embodiments. Moreover, note that the numerical
values provided are intended as illustrations of the measurement
technique. In other embodiments, the numerical values can be
modified or changed.
[0142] The foregoing description is intended to enable any person
skilled in the art to make and use the disclosure, and is provided
in the context of a particular application and its requirements.
Moreover, the foregoing descriptions of embodiments of the present
disclosure have been presented for purposes of illustration and
description only. They are not intended to be exhaustive or to
limit the present disclosure to the forms disclosed. Accordingly,
many modifications and variations will be apparent to practitioners
skilled in the art, and the general principles defined herein may
be applied to other embodiments and applications without departing
from the spirit and scope of the present disclosure. Additionally,
the discussion of the preceding embodiments is not intended to
limit the present disclosure. Thus, the present disclosure is not
intended to be limited to the embodiments shown, but is to be
accorded the widest scope consistent with the principles and
features disclosed herein.
* * * * *